Measurement

SI fundamental unitsMass; Length; Time; Temperature; Amount

SI Prefixes

1012 - Tera (T)

109 - Giga (G)

106 - Mega (M)

103 - Kilo (K)(k)

100 - Base unit

10-1 - deci (d)

10-2 - centi (c)

10-3 - milli (m)

10-6 - micro (μ)

10-9 - nano (n)

10-12 - pico (p)

10-15 - femto (f)

10-18 - atto (a)

Measurement Accuracy – How close a measurement is to the actual value Precision – How close together a group of measurements is

n.b. - Precision may also refer to how fine a particular instrument will measure. The finer an instrument, the more likely a group of measurements will be closer together. eg. graduated cylinder vs. beaker.

Uncertainty - Each instrument has a limit to its precision.

Measurements are typically reported to 1/10th of the smallest division for instruments with markings.

The last digit is estimated and the uncertainty of the measurement is given as + 0.5x the reported precision.For electronics the uncertainty is + the last displayed digit.

Measurement Because each measurement has uncertainty there is a

limit to the precision with which we can determine an answer. This precision is determined using sig figs.

What is significant? All nonzero integers Trailing zeros followed by a decimal ex: 200. = 3 sf Zero to the right of a decimal, with a number in front ex: 0.100 = 3

What is not? Leading zeros and zeros to the right of a decimal with no nonzero

number in front ex: 0.00300 = ? Sf Trailing zeros not followed by a decimal ex: 200 = 1 sf

For multiplication and division, use least # of digits For addition and subtraction, answer should match #

with least precision

Error Analysis There are two main types of errors that can effect values

Random error – equal probability of being too high or low examples: estimating last digit, experimenter’s error (technique)

Systematic error – Occurs in the same direction each time examples: incorrect calibration, defective instrument

Two common ways to present error Compared to expected value (% error) = Exp – Theo x 100

Theo Precision (deviation) – comparing a set of measurements

A good strategy for data analysis is to compare deviation with uncertainty to see if the deviation is within the uncertainty range of the measurements used to determine values, or if some other error (eg. experimenter) is involved.

Types of Deviation Average Deviation

4 general chemistry students measure the mass of a text book

1 – 2.38 kg 2 – 2.23 kg 3 – 2.07 kg 4 – 2.55 kg

a. Determine the mean

b. Determine the absolute difference between each value and the mean

c. Add the differences together

d. Divide by the total number of measurements

e. Express answer as: mean + average deviation

Percent average deviation Divide the average deviation by the mean; x 100

Express answer as: mean + % deviation

Percent (average) deviation is expressed to only 1sig fig

n.b. Calculated values for mean match the precision of the measurements used!

Types of Deviation Standard deviation

Steps a and b are the same Square each difference Use the following formula to calculate the standard deviation:

√∑(x – x)2

(n – 1)

n.b. – The precision for the mean and the deviation should match (except for % deviation).

_Be sure to express the standard deviation as a function of the mean.

Types of Deviation Practice Example:

Six groups of students each experimentally determined the thickness of the Zn layer on a piece of galvanized iron. The following values were reported (in cm.):

.00193; .00220; .00189; .00216; .00278; .00226; .00208

Express the mean value for the thickness of Zn including Average deviation % deviation Standard deviation

0.00219 +/- 0.00020 cm

0.00219 cm +/- 9%

0.00219 +/- 0.00030 cm

Dimensional Analysis A method used primarily for unit conversion.

Works by multiplying with conversion factors

Example: A pancake – eating contest was won by an individual who ate 74 pancakes in 6.0 minutes. At that pace, how many eggs would he have eaten in 1.00 hour? Assume 1 egg was used to make 8 pancakes.

Light travels at 186,000 mi/s. How many centimeters would light travel in one year? Assume 365.24 da/yr and 1 mi = 1.609 km.